NOTES. 359 where (X, /*, v] have all possible values. It is obvious that the term containing the product X/A disappears on integration, for the elements corresponding to (X, /*) and (X, - /A) destroy each other. In the same way the term containing the product \v disappears. We therefore have X2 dta tf=-p7? These may be written in the form X=—Api~, 3f= — J5/J7/, Z—-Cpf........................(4\ We notice that the constants A, B, C are functions of the ratios of the axes and are therefore the same for all similar ellipsoids. The integrals given above for A, B, C may also be written in the form ...........(5), where the integration extends over the whole surface of the ellipsoid. It easily = -du... where r is the radius vector of the bounding ellipsoid drawn from the centre as origin. The potential is seen by an easy integration to be V=^p{D-A^-B^- C£2}, where D=Jr2dw, since $pD must evidently be the potential at the centre. NOTE D, Art. 218. other laws of force. The potential of a thin homogeneous homoeoid at an internal point (fyf) when the force varies as the inverse /cth power of the distance can be found, free from all signs of integration, when K is an even integer > 2. Let pp be the surface density at any point Q, where p is the perpendicular from the centre on the tangent plane at Q. The potential is __ - 3) 2a * - - 5) where and and The general term is *J - JLJ 1 The series has 4(*-2) terms. Thus for the law of the inverse fourth power it reduces to the first term; for the law of the inverse sixth power, there are two terms and so on. At an external point P' whose coordinates are £', 9?', £', we have abc where ~aW(/c-l)(/c-3) \E' /_a/4 '*2 k'4 ^2 ~a^ d£'2 1?dtf' 1 + s Here a', 6',c' are the semi-axes of the confocal drawn through P', and e2=a/2 - a2=